Variational and numerical aspects of a system of ODEs with concave-convex nonlinerities

In this work we discuss a Hamiltonian system of ordinary differential equations under Dirichlet boundary conditions. The system of equations in consideration features a mixed (concave-convex) power nonlinearity depending on a positive parameter. We show multiplicity of nonnegative solutions of the system for a certain range of the parameter and we also discuss regularity and symmetry of nonnegative solutions of the system. Besides, we present a numerical strategy aiming at the exploration of the optimal range of for which multiplicity of solutions holds. The numerical experiments are based on the Poincar\'e-Miranda theorem and the shooting method, which have been lesser explored for systems of ODEs. Our work is motivated by the works of Ambrosetti et al. , 1994 and Moreira dos Santos, 2009.